Regularity of weak solutions for nonlinear parabolic problem with $p(x)$-growth

Abstract

In this paper, we study the nonlinear parabolic problem with $p(x)$-growth conditions in the space $W^{1,x}L^{p(x)}(Q)$, and give a regularity theorem of weak solutions for the following equation $$\frac{\partial u}{\partial t}+A(u)=0$$ where $A(u)=-\mbox{div} a(x,t,u,\nabla u)+a_0(x,t,u,\nabla u)$, $a(x,t,u,\nabla u)$ and $a_0(x,t,u,\nabla u)$ satisfy $p(x)$-growth conditions with respect to $u$ and $\nabla u$.

Keywords

• nonlinear parabolic problem
• regularity
• $w^{1
• x}l^{p(x)}(q)$ space
• $p(x)$-growth condition